Many-body localization and mobility edge in a disordered spin-$\frac{1}{2}$ Heisenberg ladder

We examine the interplay of interaction and disorder for a Heisenberg spin-1/2 ladder system with random fields. We identify many-body localized states based on the entanglement entropy scaling, where delocalized and localized states have volume and area laws, respectively. We first establish the dynamic phase transition at a critical random field strength $h_c\sim 8.5\pm 0.5$, where all energy eigenstates are localized beyond that value. Interestingly, the entanglement entropy and fluctuations of the bipartite magnetization show distinct probability distributions which characterize different phases. Furthermore, we show that for weaker $h$, energy eigenstates with higher-energy density are delocalized while states at lower-energy density are localized, which defines a mobility edge separating these two phases. With increasing disorder strength, the mobility edge moves towards higher-energy density, which drives the system to the phase of the full many-body localization.

Figure 1

(a) The ratio of entanglement entropy over the number of system sites $S/N$ for different systems from $N=6\times 2$ to $9\times 2$ at the energy density $\varepsilon =0.5$ as a function of random disorder strength $h$. Curves for different $N$ approximately cross each other around a critical $h_c=8.5\pm 0.5$. (b) The adjacent gap ratio $r$ for states with energy density $\varepsilon =0.5$ as a function of $h$. Here we see that on the small-$h$ side, $r$ approaches the Gaussian orthogonal ensemble value (0.5307) representing delocalized states, while at the larger-$h$ side, $r$ reaches the Poisson value $(2ln2-1\simeq 0.3863)$ for larger systems representing localized states. All curves cross around the critical value $h_c=8.0\pm 0.5$. (c) The ratio of the fluctuations of half-system magnetization over the number of system sites $F/N$ as a function of $h$. Curves for even (odd) $N_x$ approximately cross each other around $h_c\sim 8$–9. The standard error bars for the largest system $N=18$ are shown in (a)–(c), while other data for smaller $N$ have smaller error bars. Combining the above results, we estimate the critical point is around $h_c=8.5\pm 0.5$.

Figure 2

(a) The probability density distributions of the bipartition entanglement entropy $P\left(S\right)$ for spin system at energy density $\varepsilon =0.5$ with different disorder strengths $h=4$, 6, 8, and 10 for $N=6\times 2$ and $8\times 2$. These results illustrate that for a stronger disorder case, the distribution always has a long tail into higher-$S$ values, while for smaller $h$, the long tail is on the small-$S$ side. (b) The probability distributions $P\left(F\right)$ of the variance $F=\langle {\left(S_A^z\right)}^2\rangle -{\langle S_A^z\rangle }^2$ (in units of the square of the Planck constant ${\hslash }^2)$ of the magnetization of the half system $A$ for spin system at energy density $\varepsilon =0.5$ with different disorder strengths $h=4$, 6, 8, and 10 for $N=6\times 2$ and $8\times 2$. Error bars for all data points in (a) and (b) are comparable to the size of the symbols.

Figure 3

(a) The variance ${\left(\mathrm{\Delta }S\right)}^2=\langle S^2\rangle -{\langle S\rangle }^2$ of the entanglement entropy at energy density $\varepsilon =0.5$ for different $h$. $\mathrm{\Delta }S$ reaches the peak value at $h_p$ smaller than the identified $h_c$ for phase transition. Clearly, $h_p$ shifts to higher $h$ with the increase of $N$. The error bars are comparable to the size of the symbols.

Figure 4

The entropy of each energy eigenstate $S_i$ for low-energy eigenstates averaged over disorder configurations is shown as a function of its average energy density. $S_i/N$ curves for different system sizes $N=6\times 2,8\times 2$, and $10\times 2$ cross at a critical energy density ${\varepsilon }_c$, which separates the higher-energy delocalized ergodic states from the lower-energy MBL states. The error bars for the $N=18$ system in (c) are shown, while all other data in (a)–(c) have a typical error bar that is smaller than or around the same size as the symbols.